Observing the Pancharatnam-Berry phase on the Poincaré sphere

Heather Hill, Duke University,
Marty Cohen and John Noé
Laser Teaching Center, Stony Brook University

In 1984, Michael Berry discovered a geometric phase in quantum systems [1], which soon led to a whole new understanding of the importance of geometric phases in other fields. In particular, an optical phenomenon that had been discovered earlier by Pancharatnam [2] was realized to be a manifestation of a geometric phase. The Pancharatnam-Berry phase is the phase change that a monochromatic beam of light gains in a cyclic change in polarization. If one maps out the changes in polarization on the Poincaré sphere, this phase change is equal to half of the solid angle that the polarization trajectory traces out on the sphere.

We hope to observe and measure the Pancharatnam-Berry phase with an apparatus inspired by van Dijk et al. [3,4]. We will start with vertical linearly-polarized light (Point A on the equator of Poincaré sphere) and pass this through a quarter-wave plate to create circularly- polarized light (Point B at the north or south pole). A linear polarizer at some angle α to the vertical will then move this polarization state to another point C on the equator. Finally, a second linear polarizer will restore the light to its original state, point A. The experiment consists of comparing the relative phase of the initial and final polarization states in a Mach-Zehnder interferometer. We should observe that the fringe shift as a function of α varies in proportion to the solid angle mapped out on the Poincaré sphere.

We found a number of unspecified quarter-wave plates in the lab and tested them to see which ones were effective at 632.8 nm wavelength. With careful adjustments, the best of these produces good-quality, circularly polarized light from the beam of our linearly polarized HeNe laser. The interferometer was set up and fringes with good contrast were obtained; the fringes are susceptible to air currents, mechanical vibrations, etc, but with care are sufficiently stable when the room is quiet. Data taking will consist of recording the shift in fringe position while changing the first linear polarizer from α = 0 to other values α < 180°. The Pancharatnam-Berry phase is largest (π/2, or 1/4 of a fringe) for α = 90°, which corresponds to moving half-way around the equator of the Poincaré sphere. Unfortunately, as one approaches this angle the transmission of light in the active arm of the interferometer drops to zero. Our initial observations are a series of images of the fringe pattern with α cycled between 0, 35 and 70 degrees. These images clearly show a fringe shift of the expected magnitude. We are currently working to verify that this shift is indeed due to the geometric phase.

We thank T. van Dijk and Prof. T. D. Visser for their helpful responses to our emails. This research was supported by a grant from the National Science Foundation (Phy-0851594).

[1] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
[2] S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
[3] T. van Dijk, H.F. Schouten, W. Ubachs and T.D. Visser, Proc. SPIE 7613 (2010).
[4] T. van Dijk, personal communication.